(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Effect of high viscosity on the flow around a cylinder and around a sphere"





CO 



Eh 
< 

s 



NATIONAL ADVISORY COMMITTEE 
FOR AERONAUTICS 

TECHNICAL MEMORANDUM 1334 



THE EFFECT OF HIGH VISCOSITY ON THE FLOW AROUND 

A CYLINDER AND AROUND A SPHERE 

By F. Homann 



Translation of " Der Einfluss grosser Zahigkeit bei der 

Stromung um den Zylinder und um die Kugel," ZAMM, 

vol. 16, no. 3, June 1936. 




Washington 

June 1952 UNIVERSITY OF FLORIDA 

DOCUMENTS DEPARTMENT 

120 MARSTON SCIENCF I PPARY 

P.O. BOX 11 70 

GAINESVILLE, FL 32611-7011 USA 



^(^Il^l'-I' 



11 I I 



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 



TECHNICAL MEMORANDUM 1334 



TEE EFFECT OF HIGH VISCOSITY ON THE FLOW AROUND 
A CYLINDER AND AROUND A SPHERE*^ 
By F. Homann 



For the determination of the flow velocity one is accustomed to 
measure the impact pressure, i.e., the pressure intensity in front of 

an obstacle. In incompressible fluids the impact pressure is /v /2g 

\y } ^S,/'^ } is specific weight and v, m/sec, is velocity) if the 
influence of viscosity can be neglected. Such an influence is appreci- 
able, however, when the Reynolds number corresponding to impact tube 
radius is under about 100, and must consequently be considered, if the 
velocity determination is not to be faulty. The first investigations 

of this influence are included in the work of Miss M. Barker . In the 
following pages, experiments will be reported which determine the inten- 
sity of impact pressure on cylinders and spheres; furthermore a theory 
of the phenomenon will be developed which is in good agreement with the 
measurements . 

The research apparatus consists of an oil circulation in which the 
velocity of the oil can be varied from 0.5 centimeter per second to 
30 centimeters per second with the help of a vsine-type pump lying 
entirely in the oil. A Russian bearing oil and a mixture of this with 
fuel oil is used for the measurements. Figure 1 illustrates the test 
setup. In this is indicated: P, the pump; a, turning vanes; 
G, straightener; and V, the actual test section which possesses a 
breadth of 0.148 meter, a depth of 0.15 meter calculated from the oil 
surface, and a length of O.Jk meter. It was provided with wall ports A 
in three different places. E is an entrance section for the pump; 
D, a diffuser; the immersion heater T and the cooling coil K provide 

"Der Einfluss grosser Zahigkeit bei der Stromung um den Zylinder 
und um die Kugel", ZAMM, vol. l6, no. 3, June I936, p. 153-164. 

-^The suggestion of the present work, which was prepared in the 
Kaiser Wilhelm Institute in Gottingen, I obtained from Herr Professor 
Dr. Prandtl, to whom in this place I express my most heartfelt thanks 
for the energetic furthering of the work and the valuable suggestions 
given me for its completion. Another work in the same field is published 
in "Forshung a.d. Gebiet des Ingenieurwesens", I936, vol. 7, no. 1. 

T^. Barker - Proc. of Royal Society, 1922, Vol. A. -101, p. 4-35. 



NACA TM 133'+ 



temperature regulation. For impact pressure measurement a cylinder 
which was provided with a port was built rigidly into the test section. 
The dieuneters of the cylinders used were 1 centimeter, 1.377 centimeters, 
1-953 centimeters, and 2 'centimeters. The holes had a diameter of 
0.1 centimeter and 0.2 centimeter. Two corrections - one because of 
wall effect, the other because of finite size of hole (which originated 
with A. Thorn and first had to be checked for the measuring range under 
consideration here) - were applied to the measurements, which are illus- 
trated in figure 7. The solid curve represents the theory. 

More precise information on the test setup and the measurement 
technique are found in the work cited in footnote 1. 

In the case of the measurement of static pressure on a sphere, a 
sphere provided with a hole was affixed to a pitot tube, the sphere 
having one time a diameter of O.B centimeter, the other time 1.6 centi- 
meters. The execution of the measurements was in the same manner as in 
the case of measurement of the Barker effect. The result is shown in 
figure 8. The solid curve corresponds to the theory. 

In order to arrive at a clearer picture of the viscous flow around 

a cylinder or a sphere, the case of viscous flow against a plate was 

next calculated. The differential equation appearing in the two- 

■:> 
dimensional case has already been solved by Hiemenz-^ and will be sketched 

once more for the sake of a better understanding of the final form. The 

solution will than be used in the flow around the cylinder. 

After this, the three-dimensional flow against a plate in a fluid 
jet will be treated, to be used on the sphere. 



VISCOUS FLOW IN THE VICINITY OF A STAGNATION POINT 
(TWO-DIMENSIONAL CASE) 



Potential-flow theory gives for the velocity components in the 
neighborhood of a stagnation point, for the case of flow perpendicular 
to a plane wall (fig. 2): 



U = -ax V = ay 



-%iemenz - Dissertation, Dingl. Polyt. Journal, V.326 (I9II), 
No. 21-26. 



MCA TM 133^ 



The pressure is found from the Bernoulli equation to be 

,2 



P^ - P 2 - ■ ■ - 2 



a(u2 , v^) = P|l(x2 . y2) 

With consideration of viscosity, Hiemenz^ makes the formulation: 

u = _f(x) V = yf'(x) (1) 

^o - P = ^(^x) ^ ^) ( = ) 

The continuity equation is fulfilled; the boundary conditions read: 

for X = (that is, at the wall): u=v=0, f=f'=0 
for X = co; V = V, f = a 

In equation (2) if p^ signifies the pressure at the stagnation point, 

then F/ v = 0. From the equations of motion 
(o) 

sx ay p ax \;d^^ ^/l 
^- ^y p^y W ^yV 

one obtains as determining equations for f, x gind F/ x : 

(x) (x) 

a2 

ff« = f_ F' - Vf'» (3) 

f»2 _ ffit = a2 + vf" (k) 

with the boundary conditions above. 



4 NACA TM 1334 

Equation (k) has already been integrated by Heimenz. In order to 
make the coefficients ej^ual to unity, he sets 

f(x) = A^(0 I = ax 

From comparison of the coefficients, it follows that 

A = \A^ a =^ 

With this, equation (4) becomes 

0t2 _ ^,,^ ^ -L ^ 0,,, (5) 

The new boundary conditions read 

for |=O:^=0'=O 
for ^ = 00: 0» = 1 

The behavior of and the first two derivatives is shown in figure 4. 

We need the pressure difference between the stagnation point and 
the pressure for x = ocw For x = 00: 

^' = 1 f = a 5? = I - 0.647 

From integration of equation (3)^ F is determined to be 

^ F = Vf ' + I f 2 (6) 

If one now forms (p - p^ minus (Pq' - P')^ ^.s given by the Bernoulli 
equation, one obtains: 

(P„ - P) - (P,- - P') = if (r(,, . y^) - ^-{fj * y%„.^) 



NACA TM ISS'^ 5 

For the stagnation streamline, for which y = 0: 

(p„-p)-(p„'-P')=ef^F(^, |i-.2 

If one puts in for F the previously obtained value, there results 

(P^ - P) - (Pq' - P') = |(2Vfeo' + ^I - fco^) = PVfeo' 

For X = 00, f • = a; therefore, one obtains as a final formula: 

(Pq - p) - (Pq' - P') = Pva (6a) 

VISCOUS FLOW AT A STAGNATION POINT 
(ROTATIONALLY STOMETRIC CASE) 



For the solution of the differential equation arising, all the 
expressions, such as the equations of motion, the velocity components, 
etc., were reduced to cylindrical coordinates. 

If z, r and |3 are the coordinates (fig. 3)^ then corresponding 
to the two-dimensional case, there will apply: 



V. = -f/„^ v^ = J f(^) . (7) 



P„-P 



The continuity equation is again fulfilled; v„ = 0, since we are dealing 

with a rotationally symmetric process. These expressions stem from the 
frictionless problem of a fluid jet against a plate, where 

Vj. = ar 72 = -2az 

and 

p _ p = P|!(i,,2 , ,2) 



NACA TM 1334 



The quantity 2az in the frictionless case is replaced by f/ x in 
the viscous case. In the case at hand the equations of motion read: 



dVj, dvj, 
or dz 



Bv„ Sv„ 
^ dr ^ dz 



= .1 ^ + 
P Sr 



^a^v 



-^ dVv, V 



r X ^' r 
^ dr 



r2 az^y 



,1 ^ 
P Sz 



^S^v 



av„ a^v. 



+ V 



^ + 11:^ + 



^az 



2 r Sj 



az^ 



> (9) 



Substituting equations (7) and (8) in equation (9) gives; 



i ft2 _ f-ftf = 2a2 + vf*'* 



f f • = §:_ F» _ vf ' ' ' 



(10) 
(11) 



The boundary conditions read 



for z = 0: f = f ' = 
for z = 00: f = 2a 



If one finds f from equation (lO), one can therewith determine F 
from equation (ll). One next substitutes the transformation 



f(z) = A0(O % = az 



(12) 



into equation (10), in order to make the two coefficients equal to unity. 
This yields: 

i a^A^0'^ - a^A^00" = 2a2 + va^A^'" 



From equating the coefficients; 



12 2 2 3 
- a A = 2a = Va-^A 
2 



NACA TM 133^ 7 

A = 2^ ^ =^ (13) 

From equation (lO) vrith equations (12) and equations (I3) there results 
the final differential equation: 

(3»'« + 25;>0" - 52'^ + 1 = ilk) 

with the boundary conditions: 

I 

for 1=0: = 0' = 
for I = 00: 0« = 1 

The differential equation (l4), just as Hiemenz', is no longer 
elementarily integrable. Its solution was obtained, accordingly, 
through a power series development from zero: 

^ = a^ + a^^l + agl + . . . + a^^^ (15) 



By the method of undetermined coefficients, a^^ can be determined. 

Since, however, one boundary condition lies at infinity, one coefficient 
remains undetermined; and in fact it turns out to be ao. From the 

recursion formulas 

2 3 
P = a.^ + a-j^l + a I + a^l"^ + . . . 

0' = a + 2a2| + 3a^| + . . . 

0" = 2a2 + 2 X 3a3l + 3 X 4a. |^ + . . . 



result as coefficients: 



8 NACA TM 13324- 

a^ = a^ = 

a2 = for the present undetermined 

a^ = -O.I6666T 

a^ = 

a^ = 

a^ = 0.555556 X 10" 2 ag 

a^ = -0.396825 X 10" 3 
ag = 

a^ = -0.41+0917 X 10" 3 a.^ 
^10 = 0.793651 X 10"^ ag 
^11 = -0.360750 X 10-5 
^12 = 0.371+111 X 10"^ B.^ 
a-j^2 = -0.114597 X 10"^ a.^ 
a-^^ = 0.115735 X 10-5 ag 
/ ai5 = -0.301lti32 X 10"5 B.^ - 0.385784 X 10"''' 

a^g = O.I34S96 X 10~5 agS 
a^ = -0.211005 X 10"^ ag^ 

^18 " 0.224141 X 10"^ a25 + 0.157758 X 10"''' ag 
a-j^^ = -0.135546 X 10"^ ag^ - 0.415153 x lO'^ 
ago = O.3I6633 X 10"''' ag3 

agi = -0.152798 X lO-T ag^ - 0.295658 X 10"^ ag^ 
agg = 0.119505 X 10"''' ag5 + 0.199390 X 10"9 ag 
ag3 = -0.371665 X 10"^ ag^ - 0.433457 X 10"^! 
agi^ = 0.956242 X 10-9 a^T + 0.55^360 X 10-9 ag3 
a25 = -0.943031 X 10-9 ag6 - 0.462914 x 10"^^ ag2 



NACA TM 133^ 9 

In order now to be able to determine ag^ a second series develop- 
ment from infinity was set up, which was adjusted to the boundary condi- 
tion for at infinity. To this end one sets 

■ = 0o + S^i (16) 

in which 0-, corresponds to a small quantity, which one can neglect in 
the following expressions when it appears squared. ^^ is the solution 
for ^ = 00. 

0. = 0^' + 0-^' = 1 + 0i' 

since for | = oo; 

0^t = 0t = 1 0tt ^ ^^tt 0tfi = ^^... (i6a) 

The boundary condition reads 

for I = oo: « = 
Furthermore, = |. 

The integration constant is omitted, since in the following calculation 
it comes in again automatically. 

If one substitutes the above values into equation (lO), one obtains 

i^l'" + 2(^0^1" ^ ^A") - (^o'^ "" ^o'^l' + ^1'^) + 1 = (IT) 
Or if one neglects the squared terms in : 

0^'«' + 2|0^" - 20^' = (18) 



10 NACA TM 1334 

To solve this differential equation one sets 

$ = 0i' 

With this, equation (I8) gives 

$f t + 2|$» _ 2$ = (19) 

A special, not identically vanishing solution of equation (I9) is: 

If Oo is an additional solution, then 



21 dl 



'^1^2' " ^l'^2 ~ ^ 



_p2 



*2 |2|^ ^ 



This equation is directly solvable. Its solution is 

^00 T]^ 



The general solution of equation (I9) is then: 

00 T 



«D = c^o-L + CgOg = c^i + ^2^1 ^ ^"^^ 'in 



NACA TM 133^ 



11 



Since for ^ = 00, J^ ' = $ = 0, then C-j_ = 0. Therefore 



0/ = = C2I / -^ e-n^ dTi = Cgl 

-'- J 00 TJ 



ie-^"-2/'e-1^d, 



and since 



<^-^ = * d? 



03_ = C^ + Cg j 



2 pn ,,2 

■e-n - 2^ e 

<-' 00 



-^ du 



dn 



I 



Ml 2 



C3 - Cp / e-T dT] - 2C2 / Ti dTi / e-^ du (20) 



00 ^00 



The double integral becomes, according to Blasius \, 



2C. 



T\ d^ / e"^ du = 2Cr 



1 «=2 
2 



''L ^'''"^-il --^' ^^ * i ^--^^ 



With this, equation (20) becomes: 



01 - C 



I 2 



Sr^-M-^ -ft^^^)/ -^ 



dTl 



(21) 



Now: 



o| 2 r^ 2 r~ 2 r^ 2 

2 / e"^ dT) = 2 / e-^ dT) - 2 / e"^ dr] = 2 / e'T dr^ 
^00 ^o ^o ^o 



^ 



\AtJ 



r I 2 
e ■• dT) = -f= / e'T dTl - 1 



>(22) 



^. 



o 



Blasius - Dissertation, Z.F.M.u. Phys. 1908^ P- !• 



12 NACA TM 133^ 



If one substitutes equation (22) in equation (21), one can calculate (f> 
pointwise, since 



2 r^ -.2 



\/s, 



e-11 dT] 



o 



is tabulated. Therefore 



^.-i-^^-(i-^)f{i/;-^^-: 




t=^[-i^x''="'''4 



Herewith 0, 0', and 0*' are determined for the development at 
infinity, as a comparison with equations (l6) and equation (l6a) shows. 

In both developments ao, C2, and Cn appear as unknowns. If 

one now combines both solutions at the point \= \q, and determines 

that the value of the function and the first two derivatives of the 
series development at zero are equal to the corresponding values that 
one obtains from the development at infinity, then three determining 



MCA TM 133^ 



13 



.2 


o 


• 




0) 








<D 


+ 






+3 








u 








(d 








<D 


m 






fH 








^ 


o 






::< 








EH 


+ 


+ 




u 
o 

d 

•d 

0) 








0) 


( > 




1 H 1 


' ^^~^ 




rt 


1 


f ^ 


■d 
OJ 


•H 


• 
+3 






1 


1 H 1 




0) 


Ti 


1 


1 


Q) 


C 






+5 






OJ 


-p 


Q) 






(U 


OJ 


(=• 




a; 


a 






-d 


1 


-d 


o 

«U1 


■d 


& 




rH 


(U 


(D 


OJ 


c O 


<u 


H 




H 


^ 


O 


1 




rQ 


> 




^ 


§ 


(^ O 


<p 


cvil^ 




0) 




ir\ 
ir\ 


o 




o 




,Cl 






• 




5 


«ijl 


1 


-P 


M 




o 


m 


, '^'5, 


'—-— ? 


I— ;:!_> 


a 

o3 


■H 
U 




1 

II 


O 


l^CVI 


, -«, 


t^ 


u 


OJ 
m 






■| 


CM 
O 


* 


CVJ 

o 


§^ 










•Ui 


o 






> 




•N 


0) 




UJI 


II 


+^ 


o 




CVJ 


xj 


+ 






rH 


Ph 




ON 


43 




1 


on 


•H 






-4- 




r^|OJ 




OJ 


d 


9i 


ra 


MD 


^ 




OJ 


o 


^ 


^ 


OJ 


fH 


o 




o 


u/< 




-p 


l> 


• 


iH 


1 


il/l 


ITN 


to 




■H 


OJ 


rj 




1 


CVJ 


•H 


In 


M 




^ 


OJ 


Q) 


05 




O 




II 





o 


t. J ) 


in 
CVJ 


TS- 


M 


ra 


OJ 


O 


1 


' 






a 


o 


o 


u 


(U 


OJ 


X 




t>D 


•H 




<H 


o 


o 




^ 


•H 


■p 






«Ln 




-* 


O 


ra 


o3 


•\ 


■s 




II 


OJ 


•H 




^ 


o\ 


-P 


HlOJ 






-S 


bO 


a< 


H 


H 




-=t- 


+ 


> 


C 


0) 


VO 


::( 


• ^-^ 


oa 






•H 




CO 


m 


o 


■ 


g 


+^ 


t>D 


lO, 


(U 


CM 


»i/i 




O 


03 


C 


VD 


Fh 


o 


UA 
OJ 


• 




g 




• 

o 




II 


Oj 


• 




(U 


tH 




no 




m 




•\ 


-p 


^ 


II 


O 


UA 


OJ 


+ 


•■ 


H 


H 






OJ 






^ 


03 


<u 


OJ 




o 


+ 


o 


-Ql 




-p 


o3 


-d 


un 




un 




O 


q; 






in 


• 


ro 




^ 


■d 




Co 


OJ 




03 


OJ 


-P 








CIS 


• 


VD 




Ch 


OJ 




cvj 


+ 


• 


+ 


•H 


O 


-p 




o 








ra 










, 


+ 


OJ 




-P 


tiH 










(d 


•\ 


a 


o 




•\ 


• 


o 


OJ 


00 


d 






OJ 




tu\ 




• 


o 


a 




a 


• 


OJ 




fH 


o 
o 


o 






+ 


<M 




a 


03 


-P 




u 








(D 




d 




o 


o 


+ 




ra 


a 


r^ 




Ch 


u/) 






o 


o 


o 






H 


H 




^ 




ra 




ra 


03 


ci3 




o 


*\ 






d 










OJ 


(U 




o 


+ 






ra 


o 


^ 




•H 








03 


o 


EH 




■P 


o 






> 


• 






03 


cS 








o 






d 
















o^ 








O 


o 






dj 








4Ln 


-p 







11^ 



NACA IM I33J+ 



Thereby ao is determined accurately to at least five places. For the 
coefficients of the power series this yields: 



a„ = 0.658619 



^2 

^3 = 

% = 

"9 = 

^10 = 

^11 = 

^12 = 

^13 = 

hk = 



-0.166667 
0.365900 X 10"^ 
-0.396825 X 10" 3 
-0.191261 X 10" 3 
0.52271^ X 10"^ 
-0.360750 X 10" 5 
0.106882 X 10"^ 
-0.497098 X 10"5 
0.762253 X 10"^ 



^16 
^17 
^18 
^19 
^20 
^21 
^22 

"23 

^2k 

"25 



0.385391 X 10-6 
-0.958673 X 10"''' 
0.381678 X 10"''' 

-0.259200 X 10"'^ 
0.904605 X 10"^ 
-0.252966 X 10"° 
0.161233 X 10"^ 
-0.703675 X 10"9 

0.209783 X 10"^ 
-0.970520 X 10"^° 



a^ = -0.605859 X 10 



-6 



a^ - aj^ - a^ - Bg - 



The values for ^, 0', and 0'' are shown more accurately how- 
ever, in Table I. In this case, 0'' is calculated accurately to two 
decimal places, 0' to two, and to three. With this the differ- 
ential equation (l4) is solved. 

From integration of equation (ll), one obtains 



3^ F = vf ' + i f 2 = 2av(0' + 0^) 



(23) 



NACA TM 133^ 15 

As in the plane case, one uses again the pressure difference 
between the stagnation pressure and the pressure for z = 00. For | = 0, 
equation (23) is equal to zero; for | = 00 

0' = 1 f = 2a 0=1- 0.557611 

If one now forms again /p^ - p\ minus (p^' - p'), as given by 
the Bernoulli equation, one obtains: 

(Po - p) - (Po' - p*) = p^^"' -^ i ^~' - 1 ^-') = p^^»' 

As a final formula one obtains 

(Pq - p) - (Pq* - P') = 2pva (2I+) 

STAGNATION PRESSUEE ON A CYLINDER 



For the stagnation streamline the Navier-Stokes differential equa- 
tion gives 

u^ + i^=v^+v^ (25) 

Figure 12 shows the variation of u on this streamline. The different 
behavior of u at the stagnation point from potential flow is explained 
by the influence of viscosity. If one integrates between the boundary R 
and 00^ one gets 



"k" 



"^^^pH-M-ri^-rg" 



Figure I3 shows that 



V ^ 



R 

= 
00 



16 NACA TM 13314- 

likewise Uj^ = 0, and we want to identify pp with p^ of the pre- 
ceding calculation. One obtains, therefore 

dU 2 r^ 5.2 
(Po - P) - ^ = P^ / H ''^ ^^^^ 



00 Sy 



■ R 2 

2-ii dx 

° Sy2 



is calculated approximately in that for u the value corresponding to 
the potential flow is put in. The contribution of the boundary layer 
to the integral is, in the case of not too small Reynolds number, small 
in comparison. As potential function $ of the flow around the cylinder, 
one obtains 



= 4 ^ f. 



and with it: 



^ = _u Z'- 25! + ^^ + 8y?R! _ ^Qx^ygRg ' 



For y = 0, therefore, along the stagnation streamline: 



A 2 h 
ay X* 



-r&-=^ 



Herewith equation (26) gives 

(Po-P)=^-^ (27a) 



p\]J 2pvU, 



NACA m 133^ 17 



or 



^^ = 1 . ^ (27) 

pU„2/2 ^- 



If one substitutes 7 = pg, then formula (2?) reads 

7UoV2g R^ 

where g = 9-8l meters per second . 

In order to be able to accomplish a comparison of test results with 
theory^ the "displacement thickness" (see Tolmien: Hdb. d Experimental- 
physik, V. k, 1st part, p. 262, "Grenzschichttheorie (Boundary Layer 
Theory)") on the cylinder must yet be considered in the calculation. 
Solution of the differential equation (5) yields (fig. 5): 

I* = ax* = a6* = 0.6ij-7 
where 5* is the displacement thickness. Therefore 

0.647 = 6*/| (28) 

If one compares the flow in the region nearest the stagnation point for 
the cylinder and for the flow against a plate, one obtains from equa- 
tions (27a) and (6a) 

2pvUo 2Uo 
pva = -^, a = -^ 

If one substitutes this value in equation (28), one obtains for the 
displacement thickness 

0.61+7 = \/^ S* = ^\^Ri 
VRV R 

The dependence of 6*/R on Re is shown in figure 6. 



18 NACA ™ 133ij- 



Now since in the test results Re is formed from the cylinder 
radius R, the actual effective radius is therefore (R + 5*), and equa- 
tion (27b) is altered to: 

^° " ^ = 1 + ^^ 



„ 2/_ Uo(R + 5*) 
7U0 /2g °^ ' 



With this one obtains as a final rule for the stagnation pressure on the 
cylinder 

"° ■ '- = 1 . ^ (29) 



7Uo^/2g Re + 0.il57^ 



In figure 7 "the solid curve again gives the theory, which agrees very 
well with the practice. 



STAGNATION PRESSURE ON A SPHERE 



Corresponding to a cylinder, for the stagnation streamline of a 
sphere 



au 1 ap a^u /a^^ s2u^ 

u + — -^^ = V + v( + 

ax P dx ^^ Uy2 ^^2 



If one integrates again over x from 00 to R: 



(Po-P)^^-v/;(0.^)a, ,30, 



Since 



.R .2 

o u 



dx = 0, % = 



^ 00 Sx*^ 
The integral on the right side of equation (30) one again solves by 



MCA TM 133^+ 19 



substituting for u the value for the potential flow. The potential 
function of this flow is 



* '- '""(' * S3) 



For potential flow it is further true that 

2 2? 

S u d u 9 u - 

+ — 7=; + — ^ = 



Therefore 



ox dy oz 



( ^ u S u K _ r B u , 5u 



R 



If one substitutes in the last formula the value for bu/bx given by 0, 
one obtains 

m ■ »■ ■ ^ 

Herewith equation (30) becomes 



2 

)_ 
'o " 2 R 



p„ - p . £^ . 3PVU, ,^^^j 



or 



^° " ' = 1 +. ;^ (31) 



pUoV^ 



Re 



or with 7 = pg 



''-' =1^4 (31.) 



7u//2g 



Re 



20 



NACA TM 1334 



If here one also puts the displacement thickness Into the calculation, 
one gets (since in the rotationally symmetric case I* = 0.5!)76): 



0.5576 = &*Ja/^ 



Comparison of equation {2k) and equation (jla) yields 

., - 3Uo 



2R 



0.5576 = \/^ B* 
V 2VR 



5^ ^ 0.^55 
R 



\fRe 



(32) 



It appears that the displacement thickness I'or a sphere and a cylinder 

are equal within ^ percent, although the displacement thickness in the 

case of plane flow against a plate is different from the corresponding 
three-dimensional flow. 

If one considers the displacement thickness in equation (31b), one 
obtains as a final stagnation pressure formula for a sphere 



P - P 
^o ^ 



= 1 + 



Re + 0.it55v/Se 



(33) 



The solid curve in figure 8 corresponds to the theory; the agreement 
with test results is again satisfactory. 

From the final stagnation pressure formula the dependence of the 
numerical factor c on Re can be determined, if one sets 

Pq - P ^ 1 ^ _!. 
7Uo72e ' ^ ^^ 

For the sphere there results 



e = 



6 Re 



Re + 0.i+55v^ 



(3^) 



NACA TM 133^ 21 

From Stokes' calculation one obtains for small Re: e = 3. In 
figure 9 is drawn log Re as abscissa, g as ordinate. In the region 
from about Re = 0.1 to Re = 1 the course of c is essentially dif- 
ferent, since Stokes' law describes an approximation for very small 
Reynolds number and the above law is an approximation for large Reynolds 
number. 

For the cylinder one obtains in the same fashion 

e = ^^V-^ • (35) 

Re + O.h^lVRe 

According to Lamb-^, for small Re, for which the validity of the formula 
extends to about Re = 0.5: 

^^_^e_^ Re (36) 

1.309 - ZnRe 



In figure 10 is again shown the dependence of g on log Re. Within 
the accuracy of measurement the test results here also confirm equa- 
tion (29). 

With the help of the flow against a plate it is now also possible 
to establish approximately the course of u, du/Sx, and from this p, 
on the stagnation streamline. A single curve was assumed in which, 
inside the displacement thickness &*, the magnitudes as given by the 
flow against a plate were used. From the displacement thickness on, 
which had a value of O.Oil55 in the foregoing case for Re = 100, the 
potential flow was calculated. To explain the transition from viscous 
to potential flow, I would like to go through the calculation of u as 
an example. The solution of the viscous problem u, /u has as asymptote 

the tangent to the curve U2/uq, which was determined from potential 

theory, at the point 6* = O.Oij-55 centimeter. In figure 11 this tangent 
is labelled t. The difference k between the asymptote t and u-]_ 

at the point x^ gives the deviation of viscous flow from potential 

flow at this point. Therefore to the value u-. at the point x^ was 

added the proper k. With the help of this procedure one obtains point- 
wise the transition from Un to uo. 



-^amb - "Hydrodynamics" (2nd Edition 1931; German Edition by 
Helly, p. 696, par. 3^3). 



22 NACA TM I33I1 



Su/3x was determined correspondingly; the pressure p was found 
from the equations of motion to be, in the case of the sphere: 

^ "^ =1-''A-^^^- ^—^ (37) 



7UoV2g ^00 200 ax Uo(R + x)^ 



Instead of 6 in the last term of the preceding equation (37)^ in 

the case of the cylinder one gets the factor k. Figures 12, 13, and ik 

are the results; by way of comparison the corresponding curves for the 

cylinder and the sphere are shown on one sheet. The curves are true, 

as already said, for Re = 100, in which R = 0.01 meter; Uq = 1 meter; 

2 
V = 0.0001 kilogram X second per meter was assumed. 

Naturally the last curves give only an approximation, which can be 
made essentially better through a second approximation; yet this task 
in the framework of the foregoing work would lead too far. 



SUMMARY 



In the foregoing work the stagnation pressure increase on cylinders 
and spheres brought about through the influence of large viscosity, was 
reported on. 

For the three-dimensional problem, hence the flow around a sphere, 
a differential equation was set up which corresponded to that of Heimenz, 
who had already solved the two-dimensional case. The solution was 
ascertained likewise through an approximate method. The solutions for 
the two- and for the three-dimensional case were used for the flow 
around the cylinder and sphere respectively; the formulas so obtained 
for the stagnation pressure increase stood in good agreement with the 
reported test results. Finally, a procedure to determine the velocity 
and pressure variation, as well as the variation of bu/bx on the 
stagnation streamline was shown and used on the practical case of 
Re = 100. 



Translated by D. C. Ipsen 
University of California 
Berkeley, California 



MCA TM 133^ 



23 



TABLE I 



1 





0. 


0" 


1 





0. 


r 











1.3172 


1.1+ 


0.851+6 


0. 91+76 


. 1697 


.1 


.006 U 


.1267 


1.2172 


1.5 


.9502 


.9635 


.1301 


,2 


.0250 


.2434 


1.1173 


1.6 


1.01+72 


.9762 


.0895 


.3 


.05J+8 


.3502 


1.0181 


1.7 


1.1^^53 


.9863 


.0622 


.k 


.097^ 


.l^U71 


.9200 


1.8 


I.2I+2I+ 


.9905 


.01+18 


.5 


.ii^39 


.53i^3 


.8235 


1.9 


1.31^36 


.9935 


.0276 


.6 


.2012 


.6129 


.7298 


2.0 


1.1+1+30 


.9962 


.0180 


.7 


.2659 


.680I1 


.6U00 


2.1 


1.51+13 


.9979 


.0115 


.8 


.3370 


.7i^00 


.55i+8 


2.2 


I.6I+09 


.9986 


.0073 


.9 


.^137 


.781+7 


.h7k2 


2.3 


1.71^16 


.9991 


.001+2 


1.0 


.U951 


.8352 


.U015 


2.1+ 


1.81+17 


.9995 


.0027 


1.1 


.5805 


.8712 


.3351 


2.5 


1.9I+20 


.9997 


.0016 


1.2 


.6653 


.9025 


.2760 


2.6 


2.0I+23 


.9999 


.0009 


1.3 


.7608 


.92I17 


.221+1 











2k 



NACA TM 1334 




Figure 1.- Test tunnel. 




Figure 2.- Streamline picture for flow against a plate (two-dimensional). 




Figure 3. 



NACA TM 133^ 



25 




Figure 4.- 0, 0', 0". The curves drawn out illustrate the two-dimensional 
solution, those not drawn out the three-dimensional. 




3! 

R 















04 












0.3 
02 






















0.1 



V 










^ , 




— 




' — 



20 40 60 80 
Re 



Figure 6. - Momentum thickness on a cylinder — = MiJ. and 

* ^ \|Re' 

sphere ^ = M^. 
^ \(Re 



on a 



26 



NACA TM 13 3U 



Test Point 



Pq-P , 



Re + 0.457 N Re 



4 

Pq-P , 




29 








■ 




29 


1 
• 














\ 










> 0-, 


« i 





20 40 60 80 100 120 140 

Re 



Figure 7.- Stagnation pressiire on cylinder. 



Pq-P 



2g 3 




Figure 8.- Stagnation pressure on sphere. 



NACA TM 133^ 



27 



o Test Point 
Curve i: £ = 3 (stokes solution) 



Curve n:€ 



gRg 



-3 



Re*o.45 5\jRe 









8 


o 

" 










^ T 


■S^^o^ 






i-^ 


X 










y 


^ 















-2 



I 2 

^Log Re 



Figure 9.- € for sphere. 



o Test Point 

Curve I- € = ^^ \*'^ — Re (Oseens solution) 

1.309 In Re 



Curve 


n;€ » 


Re + o.457VRe 














A - 












" ' 










r'^ 










y^ 


^n 










< 







-3 



-2 



I 2 

«• Log Re 



Figure 10.- e for cylinder. 



28 



NACA TM 133i+ 




Figure 11.- Illustration of interpolation for transition from viscous to 

potential flow. 



OJi 



0.6 



U 
"0 



04 



0.2 






0.2 0.4 0.6 OJB 1.0 1.2 1.4 

X 



Figure 12.- Velocity variation on stagnation streamline; cylinder: Curve I; 

sphere: Curve II. Re =100. 



NACA TM 133^ 



29 



200 



_du 
dx 



ISO 



100 



50 






0.1 0.2 0.3 0.4 0.5 0.6 07 

R 



Figure 13.- Variation of — on stagnation streamline; cylinder: Curve I; 

sphere: Curve U. Re - 100. 



u 



0.9 






2g 



0J9 



091 








a2 0.4 0.6 as 

X 

R 



1.0 \2 1.4 



Figure 14.- Pressure variation on stagnation streamline; cylinder: Curve I; 

sphere: Curve H. Re - 100. 



NACA-Langley - 6-11-S2 - 1000 



«-"--« O CO 



e 

o 



T3 

m o 

O 1^ 
^ < 



m 



ii > 



— o 
f^ CO 



H K 



y at w 
>^ I- H 

73 eg 01 

o <u x; 
< K S 



N CO 



g^ ^ 



CO ^ 



P U 

s < 

K 2 



3 0) 



x: > a 



s « s ^ 



^ "I iZ 3 



T-( eg CO 



^ ^riS 



(£3 1-, 

K o c ' 
Q -a JC '- 



" s; 



o t/. 
■" O 

i 33 
la 

^ SO 
CO .2 H 

2 > u 



Q £f 
Z £ 

M 
K 
M 
Q 



^ < 1) CO 



K o , 



3 -3 



hD ! 



< 5 



< 5 



.2 MO 



U „ - 

< B K 

Z Z H 



c in o 



■K o c ™ •" 13 



° .-S w S -g 



tfi a* ra 






^1 



=" o 

5 C 
be p 



p S S .2 x: 6 S 



° U <= 

01 ^ )X 

a > p 

3 C J. 



IS 



«- 

W TT 



O ^ 



! ° 



« o 
•.Sf 2 



_ 'I' 
2 S 



'-' 01 c 



« !" r; ■-- 



> n -P 
e -t-j P* 



p -a 
o 



S -p 
01 "^ 
T3 0> 



-rr P 

' 'S -^ 

; > 0) 

.^^ 

' Ql M 

- n 

■" Si 

■O 3 

g m 

■5 "> 



^S o 

<U D. C 

> X >, 

■^ OJ OJ 

CJ r- K 

0) S ^ 

&■$« 
0) c 

y, a a 
S P--P 

i e ^ 

j= o -3 

0) c 

•§;;« 

•^ ^ c 

^. C XJ 
O) — OJ 

-a « c 
:n o B 



tH 2 



q; 






^ ^ I- UH e- ^ 

z; c a, o H c 



o > 

fi rtl 

o. Z 
a 3 

- M 

:^ u 
.S <i; 





U) 


-a 


s 


"^^ 


"o 


m 


oT 


c 


a> 


c 


>> 


u 




<u 


to 


p 


K 



tH ^ <U 



QJ 



W 



rt 



C O 



. O M O 



bS> 



o 



^ o 

s > 



2 i5 S 



CTJ ^ 



< 

< 
z 



z 



g. c 






<u 


o 


SI 

P. 


U 


a 


3 


c 


T3 



a -Z o 

— rt 3 o 

«-G 6 5 
c2 5 a 

2 « x; 3 
:S > w c: 



lid Sd 



3 



i; > 






o <u 

(U CO 






■a a <u 

(U J2 
^t CO -ti 

01 0) =2 
< K S 



o < 
a z 



T3 - 

a> ^ o ^ 
-c > a 



<rj ^ 



en ; 



C OS 

2 « s - 



0) 01 * g 
N S Si 



.-t CM CT) 



ITS i-i [13 m 



o 

►J 

■ b 

tn 



O t/j 

-" o 

S OQ 



= a, 
bj §>§ 

K 3 Ki 

w g 

ffi O c 
0. i E 

<n M « . 

< ^ I 

Q -a 33 
Z .« . 

P OJ (I, 
Q hi) Ml 



_ 0) I (U 



rt :5 



■S o c « ■" Z! 



y 



•^ > I ^ ^ 



" oi "i 

^ x: 

H S - 

CD 

<; (u CO 
ri *! ''^ 

VJ T3 _, 

Ss 5 



en ► 

• w £• 

111 

: " « 
p-o c 
! c O 



? ° 
o Ji 



5 as 
19 

^ SO 

" .i2 H 

" -S tj 

S<£ 

< §" 
o -2 w 

<; « K 

Z Z H 



ti; ot — 
Z ■" 3 

S (2 >>" 



c^ nj CO 



O !r. 



N 



5 Q * 
< ci-a 



•-^3 O 

bD-— CD 

C7> 'rt -5 

^ O !> 

2 ^ S 



■" o 



OJ :;2 



x; 



o 

u 


to 

c 

s 


Q. J3 


u 
•a 


x: 

QJ 

x; 


o 
o 


a; 


3 
W 


C 

o 



■a 

<D 


T3 

C 


0) 

t-. 


x: 

a; 


in 

c 


o 
m 




> 


t/1 




o 






a> 




u 


rt 


c 
o 


T3 


j= 




o 


3 


<D 






Uh 


O" 


y} 





c 






c 


IT( 


'*-' 


o 










0) 


c 


UJ 
0) 

j:: 


c 


■3 
o 




3 

to 
to 


to 


K- 


CD 


s 


E 


a 









, — . 














• T ■0 


41 


rt 


ra 


i" 0) 


>• 


c: 




0) p. c 




hr 








c 

'to 


specti 
with e 
at Re 






Qj 

s 


, re 
red 

nne] 




c 


TI 


"\ a a 


g 




S S " 


H 


a> 


% 


^ 




j= .-J 


c 



rH CO 


C7 


<N t- 


rt r-C 


CO 


ca CQ 


=.;! S 


f-4 


















o- o* 


>» 




c c 


w 




■g "-g 








^ < 




H g H.a 






J= M j= n 


>: 


>. 


t< >. h c 


" OJ 


;-i 


en T3 eu q; 


> ■-* 





a; 1) x: 


"O 


OJ 


w t^ en *i 






dj flj OJ ™ 


t. CQ H cc; < a S 



e-- 



CO ^ 



3 a) 



c: H 

o < 
X Z 



x; > a 



■-■ *^ ^ CO 

tH S: :3 CO 

■g »; S2 

S « a; § 



»-H eg CO 



m I-; ^ 1=^ 



•a ra 



•71 -- •- O -K 



< 



z 

E 



■as:'" 



^ o *J w 



? "J m 

o *- i3 
- •= S 



I 

■ h 

en 

" W 
^ K 

Sh 

ti o 

O Uj 

- o 



li 

U 33 

^ SO 
CO « H 
"SO 

< g" 

u 2 w 

< eo 33 
Z Z f-i 



Wg'g 

K 3 H^ 

w E 

W o c I 

a i c 

CO C/-J ec 

< u § 

Q T3 ffi 

Z .- . 

D a; (1. 



en 5 -^ ■ 



■- c ■ 



i. — I x: 



CO SiS 

'^ x; 



en -t-i ^ 
ec 11 

o 

5 ^ b J! 



g-d 



C en 
o ai 
•^ en 

S3 

ed en 



CO 



bU 



Q 

Z 
< 

K 
U 
Q 

Z 

J 
>< ™ 

^ en 
< 2 
Qc 



en 
bd 

3 

^« 



< en < 

OH eji 

5 en eu 

. fed) "^ 

^ C - 



-2 l^ TJ c "^ <i* ■ 



& £ o 






0) en 

£ c 
en 

= .S 

5? 

£ ai 
*" ti 



I en 

ti> ^ — ' 

— 00 

ai a IS 

> X >, 

7; 01 en 

&■? « 

a) c 

"'« S 
a* ax; 

S g " 

-co-: 



en 



T3 en 13 ^H 



en 



s . 






r-.p o 

en «-H c 

bo M-H to 

.2 'C -^ 

I T3 x: • 

, u > 

i d " - 

" 05 "^ rt5 



^ £ 3 eu c "' 
o en X! o -; 
r^ ^ en -*•' — * en 



^ C33 ■— 

— eg aj 



O ^ 



s s-g 

in o ai 
°^ i-. S 



„ o Q' (i, 13" 'en 

t; S c ai c 

o en o . — 9i 

a Mr; en en E 

ai S « ai -r; •- 

^ en c: J-" c "^ 






-*-' a en > c ^ 
en en <S ^ C a 



g-g 

OJ "^ 

> OJ 

o x: 

■O 3 

£ en 

■£ en 



en en 

t. c; T3 

(D ■-' en 

T3 en c 

■So 3 

£■" o 

T3 £ i2 

3 en 3 
o ii en 
tH -^ en 



& 



en 



o _ 

:: en c 

^ ^ ai 

o H S 



w CO ^ 



r^ CO t-H 



, ^ ^ ej CO 



f-t >-H CJ CO 



3— ^ECla)£.en2. 






■2 



' C!- Ed aT 2- aT 5 



■"" 3 " 



ej en 
^ > 



to o 

o a> 

o < 



E-- 



QJ (J 

^E 



en en 



I CO ^ 



5 (k 



^ — x: > a 



i< o 



u 


en 


•n 


ffl 


en 




<" 





en 


e-i 


en 


x: 


E 





en 


en 


t, 


in 




x: 


en 


a; 


a> 




n 


•< 


H « < K 


S K 


Z 



u, ^ :S CO 

-g^ss 

ai rt aj § 
N ^ S 1? 



g- 3 t. 

E o ai 

(J i/J 

iS > ' 

i-" i' in 

>^ >';S 

° 5 o 



O) to 



N CO 



O QJ O 

<U to (-. 

^ OJ O) 

H CC < 



QJ rt 



CO 

c H 
en < 

o < 
K Z 



I 3 . ,0 

ai ^ o ^ 
^•3 c I 

S -S '" 
en 2 CO in 

i, E- - 

3 eu . . 
■*- 2 > a 

^ I rS S 
•gaiS2 

ai rt ai § 
N S S 1? 



^H CM CO 



m i-H 33 t-^ 



^^ eg CO 



uo HH 33 33 



O 

J 

■ t^ 
en 

" W 
a* ^ 

:h 

O I/J 

- O 
en cj 
a; w 



la 

U 33 

^ SO 
co.Mh 
" -S "-^ 

s<£ 

^ ",^ 
< g" 

o -S w 
<: M s 
z z E- 



E 

= eu 
bo c 
C 3 
3 l-= 



en I en 

m X S3 -c: — 



« 

33 o c 

a i c 

m i/j en 

< ^ I 

^ ai ° 

Q TJ 33 

Z .- . 

33 a; b. 

o-° 

Q bD 5P 

Z j= sd 
-»• en "^ 

tsi 2 

t- -a 

en c 

en C 

en 3 

°^ 

bD § 

en 



en :5 



en c . 
t» 3 ,~ 



CO g'2 



■*-' re t^ rt ''^ 



"^ ■*-' Xl bD_^ 



?, .5^ " E 



CO 



< en CO 

U;t^ ejs 
^ —1 

5 rt en 
S .3 



c 2-= J- 





QJ 


tn 


rt 




U 




r 














Q. 




0) 


to 


W 




(1) 


^ 




bD 


E 


!-• 


N 









C 


C 





I en 

.S' a! o 
en a c 

> X >, 

Tl ai en 

o r: CC 

ai t: ^ 



en 



en 



Ox: ,E 



1 /-, V r- "^ 



K 
W 
Q 
Z 

o;a£ 
s s >> 



boi- 



o 

l-en en ^ 

' " S S 

i' £ o S 

; "S "S ai 



a J3 

a; .2 

en- 



53 



2 t- 
T3 -C 



CD 



J; £ 3 m c 
o en x: o 
>; »*? en -— — ' 



a 



en 



^! 

a o 

' en "^ 

! en X 
. T3 - 

o 



N 



Q ai 



"s s-g 

2 tc g 



a bD:;J en 



— ai 
.2 S 



I en S 



ai „ 

i^ ,2 

en 

"^ 2; 

■O 3 

o 5 

£ en 

■5 ai 

E a 



ai c 

S « « 
£ ax; 

fe E " 
■=0-3 

D. " O 

aj ti; 

en ,„ C3 

T3 -r3 

en en 

I-. C 13 

en ■-■ OJ 

T3 en c 

;§ o 3 

o § o 

T3 J= i2 



e_> 

< 



3 
en 



I 

■ h 
en 
o M 

^ 33 
gH 

° 1^ 

L. o 

a" .^ 

^H 
o t/. 
- O 
ai cj 
a; w 



U 33 
SO 

.2h 
•S o 

I <: 



o '- 2 

a ai = 
o H c 






< 

O £ W I 

< tn 33 I 

Z Z H • 



W c 3 
K 3 H^ 
W g 

33 o c 
t/> caj en 

< t, 6 

/-^ ai S 
Q -a 33 

Z .- . 
D a; (I, 

Q Sf 3 

Z x: s^ 

< en " 

Q en E 

Z 
»-i 
■J 
f" ^ 

< 5-S 

q;h£ 
s s >> 

3 . N 



en £ 



^ x; 
•a ■" 



n en 
o en 






° .-a « E •o 



£ o aj ■ 
c .2 -5 



S rt a 
H S - 

CD 

<; en CO 

< 3 

.5^ tn 0) 



iS .3= T3 C 



en tJ 

2;! 



« "^5 rt 

CJ c '-* 

'^ bD — 

OJ OJ ca 

S "g 

. ^E 

c c ? 



en 
T3 

i" 0) o 

11 a 3 
> X >> 
■r; in en 



m : 
a-- 
en 5 



en 



ai c 
•- ,„" '^ 3 



S.5 

O T3 



be S 



ra w CO 
"..S o 

bfl'a CO 
2 fc. '- 
'■g> 
eg-S.- 

= m O 01 

3 .-. c: s 



g-£5 , 
g s ° 2: " 

« „ a; £ .2 

^5 3 01 C 

o en x: o 
y^ i— en *^ "- 



^ o "■(n cr 



01 x2 








_^ 


"rtl 


CO 


0] 


C 




C 


hr 




Q) 


m 




u 








CO 


L^J 





QJ ■ - 

E 6 5! 

-r-l (-. I_^ 

QJ 

^T3 OJ 1 

^^£ ■ 

cos 

« > ai 
2 01 x: 

■t-i OJ 1-" 

en j; o 

s «■" 

ai m 
T3 O ,/; 

t £ en 
o aJ ai 



s " « 

J' ax; 

£ E " 
x: o -3 
p. " o 



aj 



c 


S 


(Tt 








CJ 

■a 

c 


CO 

3 



CO 


3 




3 







C/J 








cu 


to 


Ih 


ii 



OJ 


« 



2 



^ •= o 
o H E 



> 01 -a 






; i, t- 



c cj 
O _ 



< 
< 

z 



o o 

o ri 



bH 



u 



2 t'J 



S > u 
x « f? 



S a 



3 

en 


T3 


O 
? 


u 


n 


M 


u 




nJ 


XI 


OJ 


C 


3 


3 





nJ 3 O 

^^2 



° 5 



■& 



< 

< 
z 



cj w 

o > 



<" 6 



M xj 



c= -o S > m c 



H 

<; 

CJ 

< 
z 



-a i 


r- QJ 




inally 
e pre 
ree-s 


6°== 




So 
h <u 






"S 




■ -o ^ 


C CJ 




o c i: 


o _ 




o « ^ 


'^ rt 




tely 1 
ocity 
tion 


c ■- 
but) 

2 ^ 

Is 




2 "3 .2 
£ > C 




X « 2 
o X •• 




§i 




t. -- 0) 




Q. 0, x; 




D. r- *-' 






o c « 


li 




4-> u _ 


T3 W 




^^1 

0) -O * 


rt 3 


o 


i- 


o 


ti o " 




ID 


s2i§ 


5 S 


XI 
S 


3 3 2 


(U JS 


3 


G -a C 


> Ul 


C 



a 



< 
z 



O CO 
O " 



3 S S 



« a. 



•i --J 0, 

I a K 



fe£>S 





O C i3 o _ 




O c« ^ 2 M 




iproximately 1 
e the velocity 
he variation o 
on the stagna 
or the practic 












a c ^^ *i **" 




" -2 M S 73 
o c n .S 0) 


eo 


■St- . T3 w 


*-• 


bers up 
' to dete 
as well 
city gra 
m and u 


s 


< 


< 

z 


num' 
dure 
tion, 
velo 
shov 







1 

2 
















i 'C 








O « M 








O > M ^ 








nally, a pr 
pressure 
ee-stream 
eamline, i 
of ReynoL 








;-. a u '-' V 




a 




t^ £ ■" m "1 

Ql Cfl 




3 




o c ii o _ 




ti 










iroxlmately 10 
the velocity a 
e variation of 
3n the stagnati 
r the practical 




< 






u 






< 






: 










g. 01 J= o 




« 




Q, C ■" -S "" 




V 


C3 


'^ S M g TD 
O H cd ^ (U 




2 


eo 


■»j t, _ -a w 




e 

t 

1 




rs up 
dete 
s well 
ty gra 
and u 


d 
o 


1 
§• 


<! 


11) i ca .^ c 
.Q - CJ s 


S 


^ 


E s: g 5 1 

3 3 2 Q> £ 


s 

3 


u 


Z 


c -a i; > m 


C 



•< 
u 

■< 
z 






u < 



e^-S 



y] >i Sh >i (- 



^ X O JZ 



y-t CSJ CO 



Lh 


rt T3 rt 


Oi 


o 


OJ O 0) 


x: 


a> 


W I-. w 




JZ 


O) 0) <u 




H 


ffi < cc 


:^ 



■g N 55 t. B 

o < ti> rt 



^5S 



^ § 



o 

u O 

"J ^ 

O O; 

•" O 



ia 

^ SO 
" .2 H 

" .S i-^ 
I^ 5 w 

O '5 w 



hO c 



u 

K 

a 

X 3 c 

CO C/-J ^ 

< ^ § 

Q -a K 

Z .- . 

Q Sf 3 

Z's: iii 

N <" 

I, ^ 



rt 5 






w 3 



CO p CO 

^ ji: 
S rt a 
H § - 

< <D CO 

< c 

■^ a <u 
S S § 

01 3 

. bx) ^ 

Xi a ~ 

rt rt CO 

^ A 



r; c 13 „ o " 
■g o c « ■" z; 



2.tfT3 C 



_-, ^ M 



c 


o 


OJ 


n 


hr 


^^ 


> 


X 










w 


c 
o 


a; 


jr 




w 




> 




0) 






C 


XJ 


(\) 


« 



T3 41 

I 1 



ii£ 



& § o 



3 .5 

O XI 



o 



CD 



< rt ►Ju. 
Z 2 H 



X K 



Q 4> 



<ii-6 



3 rt i, -- 

3 d " - 

- Qi -•-' -^ 

■■ «; c -= 

_ <5 c o 



o 

^5 3 01 C 











o 


T1 


01 


C 

rt 


n, 





o 
■3 


^ 


> 


aj 


CIJ 


j= 


T3 


*^ 






^©"•t^kcrcojilio 



rt O . — . 
, bD t:! w rt 

*; ^ i: -t: : 



QJ £ 

x: o ' 
Q. o 

^ QJ 

a rt 

W 01 

t. c ■ 
TJ rt 

.5 S ■ 
— . o 

&s 

T3 j: 
c ■" 

§i2 



rt E 

S £ 



rt 



W 



O ^ 






»-H .-H U CO 



'^ S^ 1^^ oT^, 



"ii 




—■ 


3 "^ 3 ^ 








o" cr 




wi 
3 (h 




echni 

cs 

echni 

s 

itz 

334 


J;J < 




^ e ^ -5 £ ; 

j; « x: M ,2 









a 


> In 


>. 


c u 2 c H 






^ 


rt T3 rt 01 s <: 


•■* 


S ■" 


n 


oj 0) J3 2 t , 
w I-. en " E M 




"D 


0) 


b^ 


.i5oxa)4i0)™o<; 
(mOQHBcKKSKZ 



rt rt 5^ irt 

tH S -"■ '- 

"3 01 . . 

rt ■" J= > Q, 

**-* rt - - 

0) « S 3 

N S St? 



rt CM CO 



.«• rtSS 



o 

1-1 

■ (l< 

0) 

y W 

t. o 
11 .^ 

t^ 

O Kj 

- O 
0) o 



ia 

O K 

^ oO 
" .2 H 

2 .s u 

< g" 

Q .a M 

< « ac 
z z H 



i! c 



m (/J rt 

< ^ g 
Q -a tK 
Z .- . 

t) CU In 

< s-s 



Q be 



Z s: iai 

<; ra " 
N 5; 



3 2 



m B 
w 3 

Ml S 



5 Q 01 



C X! 

rt c . 

1-. 3 ^ 
rf rt I 

CO £ "2 

5 rt d 

H S - 

CD 
< 01 CO 

u^2 

.5 « 01 

6 s 3 

. hD '^ 
^ C - 
re rt c^ 

*^--.3 O 

a, ^ c 

bB'^H CO 

3 j; -- 

•c -c • 
cj > 

'-H <*H <5 



__. Q> I 01 O 

rt £ ^ £ ■:;; " 

tj -O rt 

vr C T3 „ o " 



5 ° rt 3 

■5 w .s 2 



m 11 

■g o 
o S 



O 4) 

g 3 

bo — 

rt rt 



£ 3 
01 



rt 0) o 

01 a. c 
> X >. 

rt 01 QJ 

01 i: ^ 



rt j; CJ 

i3 .Sf-c c 



rt >- N 



c -a 



6 S 



& £ o 



n 



-£ 



J T3 a) 

01 c 

S « « 

£ S " 

j: o rt 

" 0. ° 

■2 b « 



^ ~ rt 



' r> * C/J -^ ,11 



£ 3 01 c -' 
u] x: o — ' 



a. 6 

:2t 



u o 



bO-^ 



rt <"? 



rt a' 
rt S . 



r- F* C^ 
r. C rt 



c -a 
o 



•a .5 

W OJ 

tH 3 T3 

* - 5i 

T3 rt 3 

;§ o B 

: -a ^ f" 

•5 0) 

0) rt 



3 S 






, a* =* S 



_ 0) c 

rt -c S 



rt CO ^ 



^^ ^^ V c^ 



SiCiE— ai2. aTSi 



i5 > 



rt o 
b. CQ 



01 CJ 
He 



O) W Ih 

H K < 



w =^ 









rt -. "^ 




s^^ 


rt 


2 rt < 


fO 


£ S 


01 


rt g < 


ffi 


sSz 



■a - 

C CO ^ 



rt rt (o in 

s- S '-' ^ 

■3 01 . . 

'" S " °- 

T. S :3 S" 

■Ssss 

QJ rt S 3 



^H eg CO 



in trt rt rt 



rt CO ^ 



rt rt a CO 



3^^ Sci "J^- aT* 



H < 



H 3 



OJ CO 



5 > 






rt , 
u c 

I- rt T3 
O CD O 
OJ W J-. 

H a: < 



^ CO 

■■ti en 

U .-I 



si -" 

rt rt CO in 

t, grt rt 



C S: 



1 rt •a ■ 



K S 



rt < 

S '-': 

o < 0) i^ a; 3 

K ZN ? S^ 



^^ CM CO 



in trt rt rt 





Id 


3 ^ ra 




ti y 


S 2 


u 


M K 


c H 


Z3 


mO- 


i 3 


rt 

c 


Z< 


Uj rt ^ 

t, C CO 
S CO 


^ 


Q 


•S X- 


< 


to 


I'^H 



<1> 

a 

s 





Br Zahigkeit 

die Kugel). 

tab. (NACA 


U X Q 


s s . 


>.fa2 


- ^. 2 


■a- 
CO ■> 

rt T3 


rtl 


« £^ 

3 a> . 

.- — CM 


< S 


N ■ 


D rt 


w 


U CM 


< rt 


KK 


Q 01 05 

-rt TJ rt 


Z Z £-< 



rt o; 
^ 3 

bJD '^ 



■Z 3 ■ 



rt rt J= 

nj 1 -^ 

P CO T3 

a; rt 5 



1 £ii o 



2 1^1 

3 S 



s.S 

O T3 



0; 3 C ' rt 

rt 0) ^ 

C rt rt 01 S^ 



rt 01 
^£ 

O rt 

£ ° 

^ 3 

rt o 

t< rt C '^ 

O rt O . rt 

a, bi) rt t/i nj 

a; -rt rt ai rt 

t- u) 3 |r; 3 

uj a; ho aj ai 

x; > rt -g t, 



-£ 



c w 
o <u 

— ( ;^ w 

rt rt ™ 

o c w 
QJ rt rt 

s -g 

'£ £ 3 
l-i ■" 01 

• g'S 

C G ^ 

Qj "- ' 
C P '^ 

c p a> 

O) 0) 2 

w -^ £ 

-o t- £ 
g °-o 

.2 rt 3 

— . a o 

o i 
"•35 
« > a) 
c .S £ 
° t 

' « 2; o 
S rt '" 

0) n, 

P " (- 

■a ° w 

1 £ w 
o oj aj 
S 3 '- 

rt C D. 



rt OJ I 

<U Q. 1 

> X ; 

■■^ tu ( 

CJ £ C 

01 rt ^ 



£ e ' 



TJ 


u 


C 


(XI 


rt 


-0 




0) 


u 


c 



"O rt 



-ox:" 



3 


fO 










rt 


r/l 


■i 


a» 









a> 




x: 





H 



& 



< 
u 



O 

■ lb 
en 

u H 

^ X 

b. o 
aJ .^ 

tt 

c/- 
- O 

01 (J 

li 

U X 

^ oO 

CO .2 H 
rt > U 

S ** fc 

< g" 

O s w 

■< C5 X 
Z Z H 



01 

K 3 H^ 

X o c 

tn i/j rt 

< ^ I 

/.^ ai ° 

Q T3 X 

Z rt . 



p 


01 


(xh 









K 


,^ 


^^ 


< 
Q 


0; 

bn 


bD 


Z 


x: 


rf: 


■a" 


rt 






M 




K 
U 
Q 


01 
rn 


T3 
£ 


Z 


en 


3 


1 1 


u 


XJ 


>H 


bB 


3 


u 


en 

in 


t. 


< 


3 


■O 


qT! 


c 


z 


W 


N 





u 




K 







^ ra I 
CO £ in 

S rt d 



CD 



<ii-a 



< 01 CO 

u^2 

< 3 

.^ rt 01 

£ s § 

a) 3 
. hD*"* 

.3 3. 

rt re CO 



yj trt C 
bfirt CO 

S E - 

OS ■'*• •* 

S So 

•n o 01 

C3> I. rt 

rtl VH •^ 



c in 
o QJ 
rt m 



■g O 3 " -" 

tH ^ rt - yj 

£ ° -S 3 D. 

^ .3 tJ -r o 



3| 

Z! bD 



iS Sf-O 3 
3 ■? 3 O 



■0. rt Jif 

" S .S a 

_ O T3 3 

rt rt oj S 



g-£| 



o 
ai jq 



ai rt rt 
S "g 

h£ g 

■ 3 s 

N rt r; 
c 3 ^ 

O) ■-• ' 

S S£ 

(U QJ _2 
3^ "qj ■*-' 

g°-g 



oj a G 



<u 



5^6 



a) 3 

- tc c 

rt rt 
ax: 
S " 

O r- 



01 



'" -a .= 
w 01 

3 ■a 



■O jS 

L^ rt 

o ^ 



^-« Q. CJ 
Q O S 



QJ 

re 3 



o o-r^^ a- 

; r; c ai 

> ra o . rt 

1, bD -r^ tn rt 

' S rt a^ rt 

' M 3 i^ 3 

, 0) M ai aj 

; > rt -g iH 

• .i:: tn en rt 



« > a) 

rt (u x: 
g^rt 

■^ QJ M 

"> U o 

3 re ""^ 

ai " <u 

3 "> S, 

.S "O 3 

-D 2 en 

1 £ tn 

o 0; Q) 

S c '- 



a* „ 
T3 re 

o " o 

•ax" 
3 rt. ii 



> a) — 

o !- 2 

a 0) 3 

rt J= J! 

o H 3 



<: 
u 
<: 



o > 






M T3 






[^ £ ■" 









t. ^ at 



w 



M 



o _^ 
c r! 



OJ 



0) 



u 

•< 

2 






= <" = 

s t- s 

3 3 .2 

c -a i 



>■ « t, 

O g .Q 

° § a 

0) £ 3 

> u c 






(D g 



.2 2 



^ 


w 


^ 


>> 




cn 
1 


« 


Q. 


01 


c 




(U 




<u 


(^ 


'£ 


J= 






TD 


0) 


d 


C 




o 


« 


<4-t 


'— ' 


iK. 


o 



. 


o 


QJ 


c 


C 


>, 


1 





1-" QJ 



o _ 
'13 a 



Sa &S 



>-.ti c 



« -T; M M 





'S 01 


> 


£5 






o j: 








a. a, 


x: 


§i 






Q. c 








CO 
CO 


a .5 

o e 


a 


li 




1-H 


a. "> 


^ 


rt 3 


o 


s 


3t; 




ti-O 


o 


< 


S2 




2 t 


X3 


< 


e £ 


o 


B 


3 3 


ii j= 


3 


z 


C T3 


!w 


> to 


C 



< 
z 




■S 

<" 

u 

< 

2 



H 

< 

< 

z 



. ? S 



M T3 






« a I) " 



fe £ •" 



o o 
o z! 



a; 



d) o 

" !2 

c u 

o _ 

m u 

c •- 

2 « 



< 

< 
z 



9- d) x: " o 



T3 >* 



3.2 

■a Z 



■a m 
rt 3 d 

>. « fc, 

° i B 

> u c 



UNIVERSITY OF FLORIDA 



3 1262 08105 808 2 



UNIVERSITY OF FLORIDA 
DOCUMENTS DEPARTMENT 
120 MARSTON SCIENCE LIBRARY 
P.O. BOX 117011 
GAINESVILLE, FL 32611-7011 USA